math challenge

[narrowbandpaul]

i checked on wolfram integrator whether the integral of cos(sinx) is analytic or not.

seems not.

the function itself, ie cos(sinx) is periodic, but at first glance its integral does not exist.

any thoughts?

perhaps find the fourier series then integrate, use the power series expansions, then integrate.......

[TheThing]

The answer is, of course, 42.

[Doc]

Over my head I'm sorry to say.....

But 42 is a good guess

[themos]

You don't mean analytic, do you? Do you mean expressible in terms of elementary functions?

[themos]

Axiom/fricas which implements the Risch algorithm, says it isn't expressible in terms of elementary functions.

[themos]

But the integral definitely exists, I see no reason why it shouldn't.

If you can read tex, this is the series for the cos(sin x) and its integral

(9) -> series (cos(sin x),x=0)

$$

1 -{{1 \over 2} \ {x \sp 2}}+{{5 \over {24}} \ {x \sp 4}} -{{{37} \over

{720}} \ {x \sp 6}}+{{{457} \over {40320}} \ {x \sp 8}} -{{{389} \over

{172800}} \ {x \sp {10}}}+{O

\left(

{{x \sp {11}}}

\right)}

\leqno(12)

$$

(13) -> integrate %

$$

x -{{1 \over 6} \ {x \sp 3}}+{{1 \over {24}} \ {x \sp 5}} -{{{37} \over

{5040}} \ {x \sp 7}}+{{{457} \over {362880}} \ {x \sp 9}} -{{{389} \over

{1900800}} \ {x \sp {11}}}+{O

\left(

{{x \sp {12}}}

\right)}

\leqno(13)

$$

[Ghost66]

you lost me after on. but 42 does seem a good answer:). cool and froody answer 42:glasses1:

[George Jones]

Without success, I tried the intergral in Maple, and tried looking it up in my huge Table of Integrals (biggest book in my bookshelves).

i checked on wolfram integrator whether the integral of cos(sinx) is analytic or not.

seems not.

What do mean by "analytic"? Expressable in closed form in terms of elementary or standard special functions?

The standard definition of analytic function: f(x) is analytic if at any x, f(x) has a power series expansion that has non-zero radius of convergence.

Now, let f(x) be a function such that df/dx = cos(sinx), so f(x) is the integral that you want. (Note that this defines f(x) only up to an additive constant, as required.) Is this f(x) analytic? I really don't know, but my guess is that f(x) is analytic.

Note than since cos(sinx) is strictly positive, f(x) blows up at inifinity, but this is okay. For exmple, e^x blows up at infinity, and e^x is analytic.

the function itself, ie cos(sinx) is periodic, but at first glance its integral does not exist.

By "its integral does not exist," I think you mean "its integral cannot be expressed in terms of elementary or standard special functions." I think that the existence-uniqueness theorem for differential equations guarantees that f(x) (i.e., the integral of cos(sinx)) exits. Most functions cannot be expressed in terms of elementary or standard special functions.

In fact, most function are truly bizzarre. As an example of such bizzarreness (that has nothing to do with the problem at hand), take the fact that most continuous functions are so jagged that they are differentiable nowhere! Usually, we only work with the nice'ns!

perhaps find the fourier series then integrate, use the power series expansions, then integrate.......

These methods could work, but the interchange of limits could mess things up. If f(x) has a convergent power series expansion, it is often but not always true trhat the integral of f(x) of is the sum of the term-by-term integrals of the power series expansion. Taking any integral involves a limit (of, e.g, a Riemann sum) and summing an infinite series involve a limit. In integrating term-by-term, (integral limit)(series limit) = (series limit)(integral limit). Uniform convergence is probably sufficient for this.

Physicists often proceed, with success, on the assumption that interchange can be performed.

Sorry, I've rambled on and on without helping much.

[edit]After posting, I see that themos has already said these things.[/edit]

[Ant]

I have a towel

[themos]

MathAction FrontPage

you can use free axiom maxima or reduce software from this web interface to compute symbolic answers if you can't afford mathematica.

[Jamie]

Ok Paul here is a challenge for you-

Is there any chance you could maybe dumb it down a bit so us lesser mortals might understand what has been posted????

[Tim]

any thoughts?

yeah, but they usually involve girls

[Doc]

yeah, but they usually involve girls

Erm!! Girls or wolfram integrator

I know which one would win

[skye at night]

Yes...

Can be proven by the 'Spangles/Trebor' rules governing flange dynamics..

also..

related to Muldrew's first law of thermo-hysterics..

and thought by some to be a derivative (in a quark-mechanics sense) of Dr Oetker's theorem to calculate gravitational forces at the 'Kraft-Tofu Event Horizon...

There is much to discuss here...

Steve

[Tim]

The thing that always struck me about maths Paul, is that it would be easier to understand, if it could be taught in a way which showed its usefulness.

For instance in my hobby of woodworking, and previously PC game effect scripting, I am often working out things on calculators that I could have learned to do properly at school, but the teacher never explained why it would be useful.

So out of curiosity, and not because I have a hope in hell of understanding what you lot are on about , what use is all of this?

Cheers

TJ

[George Jones]

yeah, but they usually involve girls

Erm!! Girls or wolfram integrator

I know which one would win

"I tell them that they if occupy themselves with the study of mathematics they will find it is the best remedy against the lusts of the flesh." Thomas Mann in The Magic Fountain.

Depending on the situation, this could be good or bad.

[narrowbandpaul]

jamie, I named the thread math challenge, and I kept it short using standard mathematical terminology (a bit waffly with analytic)

clearly a math challenge is going to be difficult and if maths isnt your thing then this thread isnt suited to you.

so really cant make it dumbed down.

guys.

by analytic i was meaning expressable in terms of elementary function. even a special function would be OK. Like the airy disk being expressed in bessel functions......

but I was going for an expression in terms of regular functions like sin(x), cos(s) lnx.....

thanks for the detailed replies.....

my maths is quite good perhaps, but I didnt do pure maths, so some of this stuff I deeply fascinating but I need to read wikipedia :-(

i like to post myself hard math problems......keep the mind tickin over.

thanks all.

keep it coming

[narrowbandpaul]

unfortunately TJ, there is no use......not as far as I know.

although the bessel function of the first kind does have a term cos(sinx), thats where I first saw it.

no problem having a laugh with '42' or thoughts of girls :-)

thanks for the posts guys....

any other challenging math things that are 'fun'? :-)

[skye at night]

Dearest Paul..

May I request who appointed you some sort of 'quasi-administrator'??

If you don't want responses of any particular variety.. Then don't post something that is likely to invite them...

I hope you don't work in Science as I can assure you it is one field that still does need a sense of humour...

I have 2 Science degrees and a further engineering qualification..

But somehow I am still able to use them and conduct both my work life and life-life with 'responsibility' AND a sense of humour.

My Maths Challenge..

How many 'disagreeable' forum replies does it take before one self important, opinionated member decides to make another....

"This forum doesn't like me! I want to leave !" threat?

Steve

[Doc]

Pulls up a chair....

better then watching Eastenders

[acey]

Not much of a challenge unless you give us limits for the integration, then it can be done numerically. The indefinite integral can't be expressed in closed form.

The problem is also posed here:

Integrate cos(sin(x))

with negative results, while at this site:

https://nrich.maths.org/discus/messages/114352/115657.html?1172714262

someone wonders, equally fruitlessly, about the similar problem of integrating tan(sinx).

But thanks for making me appreciate the connection with Bessel functions. This suggests it might conceivably have some sort of application.

So the real challenge is to present an interesting real-world problem in which you might actually need to integrate cos(sinx).

[ngc2403]

This all start when i posed the question of the nature of gaussian beams to paul this morning, they have an approximation to the data made of two seperate equations one for near field the other far.

bessel functions are used for airy discs too and it this way i was hoping to find an approx equation which works near and far...

P.S. if anyone know of a working approx to a Gaussian beam near and far let me know

[narrowbandpaul]

i have wanted to integrate the bessel function for the airy disk for some time, as it can be used to calculate the airy disk.

I read that a guassian can used to approximate the function out to the first maxima, which would be easier. At least the integral of a gaussian exists......the erf function

i do enjoy sort of abstract math questions.....

perhaps you guessed that?

[themos]

Ok Paul here is a challenge for you-

Is there any chance you could maybe dumb it down a bit so us lesser mortals might understand what has been posted????

He's asked for a formula that describes the area under a curve defined by the relation y=cos(sin(x)) where x,y are cartesian coordinates.

If one doesn't exist, then the only way to calculate the area is to number-crunch it by splitting the area into very thin, almost rectangular, strips and add the little areas all up.

If one does exist, you still have to number-crunch the formula to get numbers out but you feel better about it because a formula in terms of building blocks you've seen before in other contexts (sin/cos functions, logarithms, nth-roots and the like) helps you understand the global behaviour of the thing and its relation with other formulas besides giving you quicker ways of calculating the numbers.

[SC-Pulsar]

That's a bit condescending Steve.

Also I don't understand what relevance boasting about your qualifications has.

Paul posted a serious question and is quite entitled to request serious feedback on it.

I think you perhaps took that a bit too personally.

You don't know Paul personally, so are not able to comment on his credibility of working in Science.

He seems to have a lot to offer to the scientific community from what I can see.

If only people would listen to him instead of childishly throwing sarcastic comments....